The Operator Norm on Weighted Discrete Semigroup Algebras $\ell^1(S, \omega)$

TL;DR

This paper characterizes when the operator norm on weighted semigroup algebras equals a specific weighted norm, revealing surprising equivalences and providing examples to illustrate the relationships among different norms.

Contribution

It establishes a precise condition (F-property) under which the operator norm coincides with a weighted norm on $ell^1(S, \, \omega)$, offering new insights into the structure of these algebras.

Findings

Operator norm equals a weighted norm if and only if the weight satisfies F-property.

On $ell^1(S)$, the operator norm coincides with the standard norm.

Various examples illustrate the relationships among different norms.

Abstract

Let $\omega$ be a weight on a right cancellative semigroup $S$. Let $\|\cdot\|_{\omega}$ be the weighted norm on the weighted discrete semigroup algebra $\ell^1(S, \omega)$. In this paper, we prove that the weight $\omega$ satisfies F-property if and only if the operator norm $\| \cdot \|_{\omega op}$ of $\| \cdot \|_{\omega}$ is exactly equal to another weighted norm $\| \cdot \|_{\widetilde{\omega}_1}$ [Theorem 2.5 ($iii$)]. Though its proof is elementary, the result is unexpectedly surprising. In particular, $\| \cdot \|_{1 op}$ is same as $\| \cdot \|_1$ on $\ell^1(S)$. Moreover, various examples are discussed to understand the relating among $\| \cdot \|_{\omega op}$, $\| \cdot \|_{\omega}$, and $\ell^1(S, \omega)$.